Vrftsjtryk

I am reading this book about numerical methods in physics. It has the following question:

Consider the Ising Hamiltonian defined as following $$H=-sum_ {i=1}^{N-1}
sigma_i^x sigma_ {i+1} ^x + h sum_ {i=1} ^N sigma_i^z$$

Write a program that computes the $2^N times 2^N$ matrix for
different $N$

From quantum mechanics, I know that any operator can be expressed in matrix form as follows
$$H_{rs}dot{=}langle r |H|srangle$$
where $|irangle$ are any basis.

My question is what basis can I take for this Hamiltonian? Is there any other way to write this Hamiltonian in matrix form?

For completeness I'll summarize the answer here. After a fun conversation in the comments, we saw that it will be more illuminating to write

$$H=-sum_ {i=1}^{N-1}
sigma_i^x sigma_ {i+1} ^x + h sum_ {i=1} ^N sigma_i^z $$

as

$$ H=-sum_ {i=1}^{N-1}
sigma_i^x sigma_ {i+1} ^x + hleft( sum_ {i=1} ^{N-1} 1_1 otimes cdots 1_{i-1} otimes sigma_i^z otimes 1_{i+1} otimes cdots otimes 1_{N-1} right)
+ h ( 1_1 otimes cdots otimes 1_{N-1} otimes sigma^z_N) $$

where it is understood that (to prevent clutter)

$$ sigma_i^x sigma_{i+1}^x = 1_1 otimes cdotsotimes 1_{i-1} otimes sigma^x_i otimes sigma^x_{i+1} otimes 1_{i+2} otimes cdots otimes 1_{N} $$

and the subscripts are there to denote nothing more than the position of insertion (i.e. they are all $2times 2$ identity)

We then create a matrix from the direct product of matrices $A,B$ where $[A] = m times n$ and $[B] = p times q$ matrices

$$ mathbf {A} otimes mathbf {B} ={begin{bmatrix}a_{11}mathbf {B} &cdots &a_{1n}mathbf {B} \vdots &ddots &vdots \a_{m1}mathbf {B} &cdots &a_{mn}mathbf {B} end{bmatrix}}. $$

We apply this to the $i^{th}$ term and then loop over all $i$ to achieve the desired matrix.

$textbf{Quick hint :}$

Interpret spin operators (for chain of length-N) this way (spin operators act on tensor product space) :
$$sigma_{i}^{alpha} rightarrowunderset{N_{}^{text{th}}text{order direct/kronecker product of identity and pauli matrices}}{mathbb{I}_{}^{}otimescdotsotimesunderset{i_{}^{text{th}} text{position}}{sigma_{}^{alpha}}otimescdotsotimesmathbb{I}_{}^{}}$$
which makes (for $i < j$):

$$sigma_{i}^{alpha}sigma_{j}^{beta} stackrel{i < j}{rightarrow} mathbb{I}_{}^{}otimescdotsotimesunderset{i_{}^{text{th}} text{position}}{sigma_{}^{alpha}}otimescdotsotimesunderset{j_{}^{text{th}} text{position}}{sigma_{}^{beta}}otimes cdotsotimesmathbb{I}_{}^{}$$
with standard definition of direct/Kronecker product of matrices

Note also that Kronecker product is associative.

$mathbb{I}$ is $2 times 2$ identity matrix and $sigma_{}^{}$'s are standard Pauli matrices.